Proof.

First of all, due to periodicity, we can restrict our attention to the interval0≤θ<2⁢π0θ2π0\leq\theta<2\pi. Even better, we can further restrict our attention to the interval 0≤θ≤π20θπ20\leq\theta\leq\frac{\pi}{2} for the following reasons:

1.

If an angle whose measure is θθ\theta is constructible, then so are angles whose measures are π-θπθ\pi-\theta, π+θπθ\pi+\theta, and 2⁢π-θ2πθ2\pi-\theta;

If θ∈{0,π2}θ0π2\theta\in\{0,\frac{\pi}{2}\}, then clearly an angle of measure θθ\theta is constructible, and {sin⁡θ,cos⁡θ}={0,1}θθ01\{\sin\theta,\cos\theta\}=\{0,1\}. Thus, equivalence has been established in the case that θ∈{0,π2}θ0π2\theta\in\{0,\frac{\pi}{2}\}. Therefore, we can restrict our attention even further to the interval 0<θ<π20θπ20<\theta<\frac{\pi}{2}.

Drop the perpendicular from AAA to the other ray of the angle. Since the legs of the triangle are of lengths sin⁡θθ\sin\theta and cos⁡θθ\cos\theta, both of these are constructible numbers.

....θθ\thetaAAAcos⁡θθ\cos\thetasin⁡θθ\sin\theta

Now assume that sin⁡θθ\sin\theta is a constructible number. At one endpoint of a line segment of length sin⁡θθ\sin\theta, erect the perpendicular to the line segment.

....

From the other endpoint of the given line segment, draw an arc of a circle with radius111 so that it intersects the erected perpendicular. Label this point of intersection as AAA. Connect AAA to the endpoint of the line segment which was used to draw the arc. Then an angle of measure θθ\theta and a line segment of length cos⁡θθ\cos\theta have been constructed.

....θθ\thetacos⁡θθ\cos\thetaAAA

A similar procedure can be used given that cos⁡θθ\cos\theta is a constructible number to prove the other two statements.
∎

Note that, if cos⁡θ≠0θ0\cos\theta\neq 0, then any of the three statements thus implies that tan⁡θθ\tan\theta is a constructible number. Moreover, if tan⁡θθ\tan\theta is constructible, then a right triangle having a leg of length 111 and another leg of length tan⁡θθ\tan\theta is constructible, which implies that the three listed conditions are true.